This is the 3407th most frequent Japanese word.
泳い
"泳い" is the stem form of the verb "泳ぐ," meaning "to swim."
泳いでいます is the -teiru form of 泳ぐ (oyogu, 'to swim'), indicating a continuous action happening in the present.
私は川で泳いでいます。
I am swimming in the river.
泳いだ is the ta-form of 泳ぐ, used to express a completed action in the past.
昨日、彼がプールで泳いだ。
Yesterday, he swam in the pool.
泳いでもいい is a conditional form of 泳ぐ followed by the phrase もいい, which means granting permission.